a 3^N - 2^N - related series.

[Gottfried Helms]

Hi Seqfans,

 today I came across the following series, which looks
 on a first view much interesting - but, as well, could be
 somehow trivial and incidently.

 Consider the following series:

                __ oo
                \                   1
       f(N) = -  >       ------------------------
                /_        (i*2^N - 1)((i+1)*2^N-1)
                   i=0

 This series seems always to converge to

                  1
        f(N)  = -----
                 2^N

 for instance for N=1

                   1        1       1             1
       f(1) = -  -----  - ----- - ----- - ...  = ---
                 -1*1      1*3     3*5            2

 for N = 2
                   1        1       1             1
       f(2) = -  -----  - ----- - ----- - ...  = ---
                 -1*3      3*7     7*11           4

 and so on.

 This might at least be a nice addendum to my series-
 book... ;-)

---------------------------------

 But this series has another amazing property by its partial
 sums.

 If I give another parameter m, limiting the length to a
 partial series,

                  __  m
                  \                   1
       f(N,m) = -  >       ------------------------
                  /_        (i*2^N - 1)((i+1)*2^N-1)
                     i=0

 then I *always* get

                               m*3^N - 1
    1 + (3^N - 2^N)*f(N,m)  = -----------   = g(N,m)
                               m*2^N - 1

                                   3^N
 which converges for  m->oo  to   -----   .
                                   2^N

This series stems from my studies of the open question,
whether g(N,m) will ever result in an integral power-of-2,
(which is *very* likely as it was shown in some postings in
 sci.math and sci.math.research by some considerations of
 approximability of log(3/2) by rational numbers).

This it least a surprising connection of two properties
by one series.

My primary interest in this series is to try to show, that
g(N,m) can never be an integral power of 2. (Maybe one could
even show, that g(N,m) must always be a rational, like it is
for instance analoguously with the harmonic numbers)

Does someone with more experience have any idea whether this
series could be useful for the stated problem?

Gottfried Helms